In particular the CHE that encompasses all BH perturbations in \(D=4\)  can be shown to correspond to \(N_f=3=(2,1)\)  or \(N_f=2=(1,1)\)  for the radial equation in the extremal case (a.k.a. DCHE i.e. Doubly CHE) [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. For circular fuzz-balls both radial and angular equations, coupled to one another by Carter constant, boil down to RCHE that can be shown to correspond to \(N_f=(2,0)\) , reducing to \(N_f=(0,0)\)  DRDCHE for the radial equation in the small BH limit whereby \(a_f\rightarrow 0\) .
